Question

In the figure, AP=PM=MY, PQ=1, QZ=8. Find AM?

Answer 1

Solution :

According to Tangent - secant Theorem, " when  a tangent and a secant are drawn from one single external point to a circle, then square of length of tangent segment must be equal to product of length of whole secant and the exterior portion of secant "

PQ X PZ = PA²

PQ[PQ+QZ]=PA²

1[1+8]=PA²

PA²=9

PA=3cms

given PA=PM=MY=3cms

PM=3cms

PQ+QM=3

1+QM=3

QM=2cms

now, QZ=8cm

ZM+MQ=8cm 


ZM +2=8

ZM=6cm

when two chords of circle intersect internally, then the product of length of segment are equal.

AM X MY =ZM XQM

AMX3=6X2

AM=4cm

∴AM=4cm .

Answer 2

QZ = 8 cm 
QP = 1 cm
ZP = 9 cm
In triangle AMP
AP = AQ
LM = LA
LP ( commmon )
LA = LM same to L of triangle
So AP = AM = PM
In triangle  APM and triangle ZYM
AM = MY 
ZM = PM 
LAMP = LZMY
triangle AMP = triangle ZYM
= AM = MY
AP = ZY = PM = YM
So 
AM = 1 cm